Values of a Dirichlet character (reviewed)

Complete information about a Dirichlet character $\chi\colon \Z\to\C$ of modulus $q > 1 $ is contained in the list of values $(\chi(1)=1,\ \chi(2),\ \chi(3),\ \dots,\ \chi(q-1),\ \chi(q)=0)$, each of which is a root of unity $e(r) := \exp({2 \pi i r}$), for some rational number $r$.